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I arrived at Rutgers (after an exhausting 20 minute drive). Introduced myself to material/background, played with small examples. Verified some previous calculations, proved some simple properties of reflections (graph automorphisms/involutions without fixed points on edge set), and generalized to
n-rotational symmetrical graphs. Proposed several potential generalizations and possible directions, a couple turning out to be easily solvable.
Derived some simple bounds/asymptotics, did some reading on maximum number of paths of length
k in a graph given the number of edges
m. I also did a 10-minute presentation of my project, with these
presentation slides (.pdf).
Wrote program to calculate/draw results on caterpillar paths (
Wikipedia) and cycles. For example, the following shows that the
p9 weight of the caterpillar cycle
C[10,12,14,23,25,27,29] is zero:
I started this website and did some more programming, writing a routine to calculate
pk weights on arbitrary trees/forests. The majority of the time I spent working on the
homepage tool, which will be helpful for visualizing the graphs and assigments I am working with - I have plans, for example, to introduce several new settings for some proposed generalizations. Also, I met with Dr. Choi and we discussed several revisions and new ideas.
For bridge workshops, I started working on some combinatorics problems, mostly dealing with recurrences and generating functions. For example, the well-known problem about constructing a bijection between "the number of partitions of an integer
n into distinct parts" and "the number of partitions of
n into odd parts". Also, Ethan and I worked on and solved the following
interesting recurrence problem (spoilers!): given
a(0,0) = 1, a(0,m) = a(n,0) = 0 for
m,n > 0, and
a(n,m) = ma(n-1,m) + a(n-1,m-1), find and interpret
a(n,m).
I also spent some time reading and understanding the methods of this
paper (PostScript). I'm hoping to come up with some observations about the distributions/asymptotics of
pk weights in random-ized graphs.
In the meantime, my attention is being to some other places: outside of some reading in algebra I am trying to do, I've also decided that some of these
combinatorics problems (along with
102 Combinatorial Problems, that's 202 problems at my disposal .. maybe attempt one a day?) are worth spending some time on. Also, I am starting to think about questions to pose for
PuMAC (a math competition for high school students offered at my university).
Promise I'll get to those combinatorics problems eventually ..
I met with Dr. Choi again, and we began discussing how the final paper should be like, and I started writing it. I also started looking at some related literature, now that I had more familiarity with the project; for example, I verified the solution to the problem:
for fixed l, what is the maximum number of paths of length l in a tree T with m edges. I also participated in some Project Euler races with office-mate Jonathan; he wasn't too happy when I finished my programs without telling him.
For July 4th, we went to Philadelphia where we saw an Independence Day parade, then Dr. Fiorini's house for a picnic, and finally saw fireworks at Piscataway high school. The next day there was an interesting lecture on graph crossing numbers (related to planarity). During the weekend, we went to Princeton to go kayaking; Faiz and I never managed to get the kayak going straight unfortunately.
Writing a paper is turning out to be more difficult than I had originally imagined.
I worked on and presented my
final presentation (pdf). Unfortunately with 10 minutes I did not have much opportunity other than to re-introduce the problem and mention one direction. I also wrote a summary report of my research and experience here at DIMACS (basically my research paper after removing all of the proofs). However, I have yet to submit that since I still have the Prague (yay!) component to write.
Also, Harry Potter! Pretty much as good as we expected it to be. Followed by a trip to Red Lobster the next day, Sinan ordered a whole lobster and played around with it for a while.
Just arrived in Prague after the worst series of delays I have ever encountered. After 26 hours without sleep, we were brought to the Charles University mathematics department, where we had a lecture on different representations of planar graphs (fortunately it was more than interesting enough for me to pay attention too).
The next day, we listened to a talk by Samal on probabilistic methods in graph theory (slightly related to the REU project I am working on). We received a problem set to walk on: see
brief notes.
Have had lots of fun, and saw many sights: many pictures to come soon. During the weekdays the Czech students brought us throughout Prague. On the weekend, we went on a 10-mile hike winding through several quarries that ended in Karlstejn castle.
A panoramic view of the Vltava river during dusk from the Most Legil bridge, which is crossed by the 22 tram line.
The Midsummer Combinatorial Workshop occurred this week. Lots of interesting talks happened this week, and it was unfortuante that I missed a few of them due to oversleeping - I spent a lot of time this week working on a presentation myself! The slides are
here. They are a tiny bit more involved and focused then the presentation I gave during the program.
More extensive notes.
On the weekend, we went to Hluboka castle, which was quite elegant and beautiful.
Leaving Prague :( I finished up my DIMATIA report. The last two weeks have been amazing, to say the least.
Some pictures will be going up on Picasa soon.